AP Statistics Exit Tickets, by Unit

What a two minute check is actually for

A unit test tells you whether a student can carry out a full procedure under exam conditions, days or weeks after the lesson that taught it. An exit ticket tells you something narrower and more immediate: whether the one sentence or one piece of reasoning you just spent ten minutes on actually stuck, before the class walks out the door. The two are not competing tools. A test that catches a missing idea a month later is too late to reteach it before the exam; a ticket that catches it in the last two minutes of class means tomorrow's warm up can fix it.

The thirty tickets below are built around that narrow goal. Each one names a single AP Statistics topic, gives a short prompt a student can answer in about two minutes without a calculator, and states the one line a correct answer has to include. That one line is the entire grading key. A student who writes three rambling sentences that happen to contain the required phrase passes the ticket; a student who writes one tidy, confident sentence that is missing it does not. Holding the line at one required phrase, instead of grading for a full explanation, is what keeps a ticket a two minute check rather than a second, smaller free response question.

Tickets are grouped by unit so a five ticket run through a lesson sequence, or a single ticket dropped in right after the matching topic, both work without hunting through the list. For fuller practice on any one of these ideas once the language has landed, each unit's tickets link to the matching practice set or interactive.

How to read a stack of thirty in five minutes

Grading thirty tickets in five minutes is possible only because you are scanning for one phrase, not reading for understanding. Do these four things in order, and do not let any one ticket pull you into a longer read.

First, read the one line answer key once before touching the stack, out loud if that helps it stick, so you are pattern matching against a phrase already in your head rather than rereading the key for every paper.

Second, sort into three piles as you go: has the phrase, has part of it, missing it. Do not pause to fix grammar, judge handwriting, or reward a longer answer that circles the right idea without landing on it. A messy answer with the phrase goes in pile one; a beautifully written answer without it goes in pile three.

Third, count pile three before you do anything else with the stack. If it is under a fifth of the class, the idea landed well enough to move on as planned, and pile three becomes five names to check in on individually rather than a reason to reteach the whole room.

Fourth, if pile three is large, that becomes tomorrow's opening two minutes: the same ticket, reworded with a different number or scenario, run again before moving forward. A ticket that most of the class misses is not evidence the class is behind; it is evidence the language needs one more pass before the next topic builds on it.

Unit 1 exit tickets: exploring one-variable data and collecting data

1. 1.6, describing distributions. Prompt: a histogram of exam scores is strongly left-skewed with no outliers. Describe its shape, center, and spread in one sentence a stranger could picture. Answer must include: the skew named correctly as left (negative), with the median and IQR chosen over the mean and standard deviation because of that skew. See how to describe a distribution.
2. 1.7, summary statistics. Prompt: in a dataset the mean sits well above the median. What does that alone suggest about the shape, and which measure of center should get reported? Answer must include: a shape pulled up by a high value or right skew, and the median as the measure to report because it resists that pull.
3. 1.8, boxplots and outliers. Prompt: a data point sits 2.3 times the IQR above Q3. Does the 1.5 times IQR rule flag it as an outlier, and how do you know without drawing the box? Answer must include: yes, because 2.3 is greater than 1.5, so the point sits beyond Q3 plus 1.5 times the IQR. Practice this on boxplots and the five number summary.
4. 1.11, random sampling. Prompt: a teacher numbers 30 students and uses a random digit table to choose 10. Name the two conditions that make this a simple random sample rather than just some random students. Answer must include: every individual has an equal chance of being chosen, and every possible group of 10 has an equal chance of being the sample. See how to choose a sampling method.
5. 1.12, sampling problems. Prompt: a survey mailed to 1,000 randomly chosen households gets 40 responses back. Name the bias this creates. Answer must include: nonresponse bias, named explicitly rather than just bias in general. See how to identify the type of bias.
6. 1.13, experimental design. Prompt: a study wants to say its new fertilizer causes higher plant yield. Name the one design feature that earns the word causes. Answer must include: random assignment of the experimental units to treatment and control groups. Practice with experimental design.

Unit 2 exit tickets: probability, random variables, and probability distributions

1. 2.5, mutually exclusive versus independent. Prompt: events A and B are mutually exclusive and both have a probability above zero. Can they also be independent? Answer must include: no, because knowing A happened tells you B could not have happened, so the probability of B given A is 0, which cannot equal the probability of B. See disjoint versus independent events.
2. 2.6, conditional probability. Prompt: write the probability of A and B two different ways, once starting from the probability of A given B and once from the probability of B given A. Answer must include: P(A and B) equals P(A given B) times P(B), and also equals P(B given A) times P(A). See conditional probability, does order matter.
3. 2.3, simulation. Prompt: you estimate a probability by running 500 simulated trials instead of computing it exactly. What is your simulated result, and what is it not? Answer must include: an estimate that gets closer to the true probability with more trials, not the exact theoretical probability itself.
4. 2.10, the binomial distribution. Prompt: list the four conditions a random variable needs before you model it as binomial. Answer must include: a fixed number of trials, two outcomes per trial, a constant probability of success, and independent trials. Practice with binomial probability.
5. 2.11, the normal distribution. Prompt: a value has a z-score of negative 1.5. State in words where it sits relative to the mean. Answer must include: 1.5 standard deviations below the mean. Try the normal curve explorer, and see how to find a z-score.
6. 2.12, sampling distributions and the central limit theorem. Prompt: as sample size n grows, what happens to the shape, center, and spread of the sampling distribution of the sample mean? Answer must include: shape approaches normal, center stays at the population mean, and spread shrinks in proportion to one over the square root of n. Watch it happen in the sampling distribution and CLT visualizer; see the central limit theorem.

Unit 3 exit tickets: inference for categorical data, proportions

1. 3.3, confidence interval for a proportion. Prompt: name the three conditions that must hold before building a one-sample z-interval for a proportion. Answer must include: random sample, the 10% independence condition, and the large counts condition, meaning n times p-hat and n times one minus p-hat are both at least 10. See the conditions for inference checklist.
2. 3.6, p-values. Prompt: finish this sentence correctly, a p-value is the probability of getting a test statistic. Answer must include: as extreme as or more extreme than the one observed, assuming the null hypothesis is true, not the probability that the null hypothesis itself is true. See what does a p-value mean.
3. 3.7, carrying out a test for a proportion. Prompt: a test gives a p-value of 0.02 at alpha equal to 0.05. Write the one sentence conclusion in context, not just reject H0. Answer must include: a rejection of H0 stated using the study's own variable, for example convincing evidence the true proportion differs from the claimed value, not the bare symbols alone. Practice with the one-proportion z-test.
4. 3.8, errors when performing tests. Prompt: a test rejects H0 when H0 was actually true. Name the error, and one consequence of lowering alpha to avoid it. Answer must include: a Type I error, and that lowering alpha lowers the chance of a Type I error but raises the chance of a Type II error, lowering power. Try the Type I error, Type II error, and power visualizer.
5. 3.11, justifying a claim for two proportions. Prompt: a 95% confidence interval for the difference between two proportions runs from negative 0.02 to 0.15. Does this give convincing evidence of a difference? Answer must include: no, because the interval contains zero, so no difference is a plausible value. See when a two-proportion confidence interval contains zero.
6. 3.14, setting up a chi-square test. Prompt: state the large counts condition for a chi-square test in terms of expected counts. Answer must include: every expected cell count must be at least 5. Practice with chi-square tests.

Unit 4 exit tickets: inference for quantitative data, means

1. 4.1, sampling distributions for sample means. Prompt: why do you use a t-distribution instead of a normal distribution for inference about one mean? Answer must include: the population standard deviation is unknown and gets estimated with the sample standard deviation s, and that extra uncertainty is what the t-distribution's heavier tails account for.
2. 4.2, confidence interval for a mean. Prompt: besides x-bar and the standard error, what two numbers do you need to build a t-interval for a mean? Answer must include: the critical value t-star, found using the degrees of freedom, n minus 1. Look one up on the t-table.
3. 4.4, setting up a test for a mean. Prompt: a machine is supposed to fill bottles at a mean of 500 mL. Write the null and alternative hypotheses in symbols to test whether it is off target. Answer must include: H0, mu equals 500, and Ha, mu not equal to 500, written with mu, the population mean, not x-bar. Practice with t-tests for a mean.
4. 4.6, sampling distributions for the difference between two means. Prompt: for two independent samples, what do you add, not subtract, when finding the standard deviation of the difference in sample means? Answer must include: you add the variances of the two sampling distributions before taking the square root, even though you are finding a difference. Practice with the sampling distribution of a difference in means.
5. 4.9, setting up a two-sample test for means. Prompt: name the condition that specifically requires the two samples to be independent of each other, separate from how each one was chosen. Answer must include: the independent groups condition, named as its own condition and kept distinct from random selection or random assignment. Practice with the two-sample t-test.
6. 4.10, carrying out a two-sample test for means. Prompt: a two-sample t-test comparing two teaching methods gives t equal to 2.1 and a p-value of 0.04 at alpha equal to 0.05. State the conclusion in context. Answer must include: a rejection of H0, written using the study's own outcome and both groups, not just reject H0. Try mixed inference for means.

Unit 5 exit tickets: regression analysis

1. 5.1, scatterplots. Prompt: name the four things you must describe when reading a scatterplot, form and three more. Answer must include: direction, strength, and any unusual points or outliers. Practice with reading scatterplots.
2. 5.2, correlation. Prompt: r equals negative 0.92 for a scatterplot. Describe the direction and strength, and say what r never tells you by itself. Answer must include: a strong, negative, linear association, and that r on its own never establishes causation. See correlation versus causation.
3. 5.3, linear regression models. Prompt: a regression gives y-hat equals 50 plus 3x, where x is hours studied and y is test score. Interpret the slope in context. Answer must include: for each additional hour studied, the predicted test score increases by 3 points on average. Practice with linear regression.
4. 5.4, residuals. Prompt: a residual plot shows a clear curved pattern instead of random scatter. What does that tell you about the linear model? Answer must include: a linear model is not appropriate for this data, since the leftover curved pattern means the line missed real structure. Drag a point in the regression influential-point explorer; see residual plot.
5. 5.5, least-squares regression. Prompt: name the one point that the least-squares regression line always passes through exactly. Answer must include: the point made from the mean of x and the mean of y, x-bar comma y-bar. See does the regression line pass through the means.
6. r versus r-squared. Prompt: r-squared equals 0.81 for a regression. State the percentage of variation this explains, and why r-squared can never be negative while r can. Answer must include: 81% of the variation in y is explained by the linear relationship with x, and r-squared cannot be negative because it is a squared quantity, while r keeps the sign of the association's direction. See r versus r-squared.

Turning a bad stack into tomorrow's opener

A ticket only pays for the two minutes it costs if the missed ones actually change tomorrow's lesson. When pile three is large, resist the pull to just move a fraction of a grade and continue with the plan as written; instead, open the next class with the same question in a new scenario, same numbers or a swapped context, before the new topic that depends on it. A student who could not finish the p-value sentence today is not ready for the confidence interval and p-value comparison next week, and a five minute repair now is cheaper than reteaching both ideas tangled together later.

When pile three is small, resist the opposite pull, spending five class minutes reteaching an idea eighty percent of the room already has. Pull the handful of names aside instead, at the start of the next class or during independent work, and let the rest of the room move on. Exit tickets work as a diagnostic tool only when the response to them is proportional to what they actually found.